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User blog:Primussupremus/Another attempt at defining some type of notation.
This is my next attempt to define a notation that is effective and well defined hopefully this won't end as badly as the last one. Here I am going to give slight variations on the notation I am experimenting on to see which attempt works best. We start off by giving some symbols these will be letters that will be used in sequence in sequence to form stronger functions. The smallest of these letters is "a" this stands for alpha the first letter in the Greek alphabet the biggest is Z this stands for Zeta. We start with lowercase letters and then move on to upper case letters when they have run out. a= the first hyperoperator which is addition unless you include the successor function as being the 1st. We express our numbers like this (_,_,_) on the left goes your first input (a positive integer) in the middle goes the letter you are using and on the right goes the other letter you are using. For example (2,b,2)=2*2=4. (3,d,3)=3^^3=3^3^3=3^27=7625597484987. To make something clear the letters represent the hyper operators. Because Z represents the 52nd hyperoperator the system is limited to only 50 up arrows to fix we can have more than 3 terms in the brackets such as. (3(3,(2,b,2)3)3)= 3(3,4,3)3= 3(3,d,3)3 = 3,d,3= 7625597484987 so its two 3's with 7625597484987-2 up arrows between it I have decided to call this number biggie as it is big no matter how you look at it. We can even apply recursion to make even larger numbers for example a simple expression like (5,e,5) can become (5(5,e,5)5) to write this we write (5,e,5#p) where p is the number of times you recurse it. So (5(5,e,5)5) can also be written as (5,e,5#1) as you have only recursed it once. To better illustrate this we could continue on iterating until we get to a very large number. We will use (5(5,e,5)5) as our basis as that was the last number we defined. (5(5(5,e,5)5)5)=(5,e,5#2) (5(5(5(5,e,5)5)5)5)=(5,e,5#3).(5(5(5(5(5,e,5)5)5)5)5)=(5,e,5#4) as we continue on to even larger numbers like (5,e,5#5,e,5) we eventually reach a point where more additions to the notation are required. For example (5,e,5#5,e,5) could also be written as (5,e,5#(2)) where the 2 in parentheses represents how many times the expression on the right is represented in the complete expression. (5,e,5#5,e,5#5,e,5) now thats going to be a pretty big number but we could do better as this is only equal to (5,e,5#(3)) but what about (5,e,5#(5,e,5#5)) now thats going to be fairly big. But even this titan of a number is puny compared to this number. (5,e,5(5,e,5#(5,e,5#5))) this is where we have recursed (5,e,5) (5,e,5#(5,e,5#5)) times! If we ignore everything on the right for a moment and focus on the left we can define ways of creating larger and larger numbers without needing to write so much. (5,e,5(5,e,5#(5,e,5#5))) contains 2 occurrences of 5,e,5 on the left side of the brackets so we write it as (5,e,52#(5,e,5#5)). To better illustrate this here are some more examples: (5,e,5(5,e,5(5,e,5#(5,e,5#5))))= (5,e,53#(5,e,5#5)) see how easy it was to write We could even write something like (5,e,55,e,5#5#(5,e,5#5))or even (5,e,5[(5,e,55,e,5#5#(5,e,5#5))]#(5,e,5#5). We could continue to recurse this creating larger and larger numbers such as this titan of a number. (5,e,5[(5,e,5[(5,e,55,e,5#5#(5,e,5#5))]#(5,e,5#5)]#(5,e,5#5) we could then make this the input in the [ ]'s and continue to iterate it (5,e,5[(5,e,5[(5,e,55,e,5#5#(5,e,5#5))]#(5,e,5#5)]#(5,e,5#5) this many times. The answer we get out is called pentacositakonikallion but what if we iterated it a pentacositakonikallion times to produce an even larger number which I have decided to call a super pentacositakonikallion. Super pentacositakonikallion is going to be a very large number buts its still only the 2nd term in the sequence of numbers called the pentacositakonikallion sequence where the nth term in the sequence is produced from the iteration of the n-1th term. The next value after the super pentacositakonikallion is the ultra pentacositakonikallion. Here we have the 1st 3 terms of the sequence but what about the ultra pentacositakonikallionth term and so on. The pentacositakonikallionth term is called the megaga pentacositakonikallionth now thats going to be big but we can do better. The megaga pentacositakonikallionth term in the sequence is called the megagaga pentacositakonikallion. But what if we create a new sequence starting at megaga pentacositakonikallion and call this the 1st term in the new sequence. As megagaga pentacositakonikallion is the 2nd term you can see whats going to happen next. The 3rd term in the sequence is called megagagaga pentacositakonikallion thats going to be big but again we could do even better. The megagagaga pentacositakonikallionth term in this sequence is called googoobooga pentacositakonikallion. Let googoobooga pentacositakonikallion be the 1st term in the 3rd sequence and lexion pentacositakonikallion be the googoobooga pentacositakonikallionth term we can assume that there must exist a 4th sequence as there is a 1st,2nd and 3rd sequence. If we make lexion pentacositakonikallion the 1st term in the 4th sequence and lokien pentacositakonikallion the lexion pentacositakonikallionth term in the sequence then we can conclude that lokien pentacositakonikallion is indeed the limit to the 4th series. Using this as our basis we can define even larger sequences such as the 5th series and the lokien pentacositakonikallionth series. The limit to this latter series is a very large number called the pelavulian pentacositakonikallion number. The limit of the pelavulian pentacositakonikallionth series is a number called the goranasha pentacositakonikallion number. The limit of that series is a very large number called the laki pentacositakonikallion this number will be used to descrbe the limit to an extremely larger series of numbers. We call laki pentacositakonikallion sequence 1, sequence 2 is the next limit and sequence 3 is the limit after that and so on. We continue on until we have got to the laki pentacositakonikallionth limit this number is the number I mentioned before. This number is called the Larko pentacositakonikallion number this number is very large to put it simply it dwarfs all the other numbers I mentioned. The limit of the Larko pentacositakonikallion sequence will be the basis for an even larger sequence of numbers whose limit will dwarf all the previous numbers defined. If we define the larko pentacositakonikallion to be the 1st term in this sequence we call this the 1st limit in the larko sequence we can write this like with an l like this. L(1)=larko pentacositakonikallion and L(2) would be the next limit in the series L(3) is the next and L(4) is the next. The next huge number I am going to pay more attention to is the L(larko pentacositakonikallion)th limit term in this ridiculously large sequence of numbers. This number will be known as PTA1STa a number that can only be described as bloody huge. We then create another sequence of numbers whose limit will be PTA1STb. Start with PTA1STa and call this the first limit term of the sequence and PTA1STb the PTA1STath term in the sequence using this method of generating larger and larger numbers we can define larger and larger limits such as PTA1STc or even PTA1STz. PTA1STz is the beginning of another series of numbers called the PTA2ST series. PTA2STa is the limit is next limit after after PTA1STz and PTA2STb is the next. After the PTA2ST numbers comes the PTA3ST numbers then after these numbers come the PTA4ST numbers and so on. But what about this number PTA(PTA8ST)ST now thats going to be big because not only are we making single digit numbers the index of PTAST we are using are using PTAST numbers as the index. PTA(PTA8ST)ST is still not this series of numbers as you get even make this the index of PTAST. So the PTAST numbers are ridiculously huge in comparison to a lot of other numbers people recognize. But they can still be computed by a non oracle based Turing machine so there definitely not as powerful as many other Googological systems. Category:Blog posts